Macbeth does not attempt to kill Banquo himself because he no longer has to do his own dirty work. He has become king and can delegate such deeds to others. Being king makes Macbeth conspicuous. It would be harder for him to murder Banquo than it was to murder Duncan; but it is much easier to establish an alibi for killing Banquo or anybody else, because he can have the killing done for him while he makes an appearance in front of numerous other people. As king he could probably have ordered Banquo executed (and he might have even framed him for the murder of Duncan to justify the execution), but he probably didn't feel sufficiently secure in his new position as absolute ruler. No doubt he would like to be thought of as a benevolent monarch, but he quickly learns that this is impossible because too many people, including Banquo and Macduff, feel sure he gained the throne by the worst kind of treachery.
As far as Macbeth's motivation for killing Banquo is concerned, this seems to be thoroughly covered in his soliloquy in Act 3, Scene 1, quoted below in full.
To be thus is nothing,
But to be safely thus. Our fears in Banquo
Stick deep, and in his royalty of nature
Reigns that which would be fear'd. ’Tis much he dares,
And, to that dauntless temper of his mind,
He hath a wisdom that doth guide his valor
To act in safety. There is none but he
Whose being I do fear; and under him
My genius is rebuked, as it is said
Mark Antony's was by Caesar. He chid the sisters,
When first they put the name of King upon me,
And bade them speak to him; then prophet-like
They hail'd him father to a line of kings:
Upon my head they placed a fruitless crown(65)
And put a barren sceptre in my gripe,
Thence to be wrench'd with an unlineal hand,
No son of mine succeeding. If't be so,
For Banquo's issue have I filed my mind,
For them the gracious Duncan have I murdered,
Put rancors in the vessel of my peace
Only for them, and mine eternal jewel
Given to the common enemy of man,
To make them kings, the seed of Banquo kings!
Rather than so, come, Fate, into the list,(75)
And champion me to the utterance!
Macbeth is afraid of Banquo. He realizes that Banquo should have a strong motivation to assassinate him in order to make it possible for his own descendants to become the kings of Scotland, as promised by the three witches. Macbeth also feels sure that Banquo sees right through him and knows intuitively and deductively that Macbeth killed Duncan in order to become king. Macbeth thinks he can cheat fate by killing Banquo and thereby preventing him from producing heirs to the Scottish throne. No doubt Macbeth is already planning to have Fleance murdered along with his father.
Macbeth explains to the muderers why he can't kill Banquo himself:
...and yet I must not,
For certain friends that are both his and mine,
Whose loves I may not drop, but wail his fall
Who I myself struck down. And thence it is
That I to your assistance do make love,
Masking the business from the common eye
For sundry weighty reasons.
If I've read this correctly, he has many reasons for wanting to hide his part in this, not the least of which is that he and Banquo have mutual friends. He can not be seen murdering him; instead, he must, with Banquo's friends, weep for the loss of him. In public, at least.
He is concerned with how things appear. He can't be seen with blood on his hands. Were he to kill Banquo directly, he would lose favour with some powerful people (whose loves he may not drop). I think he is aware of how tenuous his position is, and of how much he needs the few people who remain on his side.
If he wasn't the third murderer, his decision to hire people instead of doing it himself may have something to do with his already growing guilt. No matter how ambitious you are, killing one of your good buddies would be extremely difficult. In addition, by hiring people, he cuts down on his level of involvement in these murders also - less of a chance that he gets caught and blamed for these, too.
Macbeth needs Banquo killed because Banquo is raising suspicions that Macbeth killed Duncan (which he did). Macbeth has to get rid of him before he tells that to many people. More than that, however, Macbeth is worried that the prophecy of Banquo’s sons being kings will come true. His reign will be barren if his sons do not succeed him. Macbeth hires two men to murder Banquo and Fleance. Macbeth hires the murders to kill Banquo so he can be assured of success. (Although many argue that the third murder in the scene is, in fact, Macbeth)